Convergent Sequences Of Functions Research Articles

A characterization of the uniform convergence points set of some convergent sequence of functions

Abstract We characterize the uniform convergence points set of a pointwisely convergent sequence of real-valued functions defined on a perfectly normal space. We prove that if X is a perfectly normal space which can be covered by a disjoint sequence of dense subsets and A ⊆ X, then A is the set of points of the uniform convergence for some convergent sequence (fn ) n∈ω of functions fn : X → ℝ if and only if A is Gδ -set which contains all isolated points of X. This result generalizes a theorem of Ján Borsík published in 2019.

On $\mathcal{I}$-uniform convergence of sequences of functions in 2-normed spaces

In this work, we discuss various kinds of ℐ-uniform convergence for sequences of functions and introduce the concepts of ℐ∗-uniform convergence, ℐ and ℐ∗-uniformly Cauchy sequences of functions in 2-normed spaces. Then, we show the relationships between them.

A note on convergence of sequences of functions

In this paper new types of convergence of sequences of functions, called semi-α and semi uniform, are defined and studied. Also the notion of semi exhaustiveness is introduced. This new notion describes the relation between pointwise convergence, semi-α convergence, pointwise convergence and semi uniform convergence for sequences of functions. Finally, relations between α-convergence, equal convergence and uniform equal convergence are studied.

For some boundary Value Problems in Distributions

In this paper we give a result concerning convergent sequences of functions that give convergent sequence of distributions in and find the analytic representation of the distribution obtained by their boundary values. Also, we present two examples.

A Further Study on Asymptotic Functions via Variational Analysis

We use variational analysis for studying asymptotic (recession or horizon) functions. We introduce the upper and lower asymptotic operators and study their domain and image. Moreover, we characterize their fixed points and zeros. Finally, we establish continuity properties of this operator, i.e., the convergence of asymptotic functions of convergent sequences of functions.

Approximation by random complex polynomials and random rational functions

We investigate random compact sets with random functions defined thereon, such as polynomials, rational functions, the pluricomplex Green function and the Siciak extremal function. One surprising consequence of our study is that randomness can be used to `improve' convergence for sequences of functions.

Approximation approach to periodic BVP for mixed fractional differential systems

A numerical–analytic technique is presented for approximation of solutions of coupled fractional differential equations (FDEs) with different orders of fractional derivatives and subjected to periodic boundary conditions. Convergent sequences of functions are constructed with limit functions satisfying modified FDEs and periodic conditions. They are solutions of the given periodic BVP, if the corresponding system of determined equations has a root. An example of fractional Duffing equation is also presented to illustrate the theory.

Korovkin type theorem for functions of two variables via lacunary equistatistical convergence

Aktu?lu and Gezer [1] introduced the concepts of lacunary equistatistical convergence, lacunary statistical pointwise convergence and lacunary statistical uniform convergence for sequences of functions. Recently, Kaya and G?n?l [11] proved some analogs of the Korovkin approximation theorem via lacunary equistatistical convergence by using test functions 1, x/1+x, y/1+y, (x/1+x)2 +(y/1+y)2. We apply the notion of lacunary equistatistical convergence to prove a Korovkin type approximation theorem for functions of two variables by using test functions 1, x/1?x, y/1?y, (x/1?x)2+(y/1?y)2.

Stress constrained compliance minimization by means of the small amplitude homogenization method

Under the assumption of small contrast between the elasticity tensors of two materials, we derive an algorithm based on an approximate relaxation of a problem that minimizes the compliance under a constraint on stress. Numerical results are presented for the short cantilever problem, where we see that, for a 1 to 2 contrast in Young moduli, and when compared with a configuration that only minimizes compliance, one can get up to a 46 % reduction in peak stress, while compliance increases by only 1 %. The basis of the method is the small amplitude homogenization technique derived by Allaire and Gutierrez, which relies on the use of H-measures introduced by Tartar to study the quadratic interaction of weakly convergent sequences of functions.

Hyers-Ulam Stability of Iterative Equation in the Class of Lipschitz Functions

Hyers-Ulam stability is a basic sense of stability for functional equations. In the present paper we discuss the Hyers-Ulam stability of a kind of iterative equations in the class of Lipschitz functions. By the construction of a uniformly convergent sequence of functions we prove that, for every approximate solution of such an equation, there exists an exact solution near it.

Korovkin type approximation theorems proved via [formula omitted]-statistical convergence

The concept of statistical convergence was introduced by H. Fast, and studied by various authors. Recently, by using the idea of statistical convergence, M. Balcerzak, K. Dems and A. Komisarski introduced a new type of convergence for sequences of functions called equistatistical convergence. In the present paper we introduce the concepts of αβ-statistical convergence and αβ-statistical convergence of order γ. We show that αβ-statistical convergence is a non-trivial extension of ordinary and statistical convergences. Moreover we show that αβ-statistical convergence includes statistical convergence, λ-statistical convergence, and lacunary statistical convergence. We also introduce the concept of αβ-equistatistical convergence which is a non-trivial extension of equistatistical convergence. Moreover, we prove that αβ-equistatistical convergence lies between αβ-statistical pointwise convergence and αβ-statistical uniform convergence. Finally we prove Korovkin type approximation theorems via αβ-statistical uniform convergence of order γ and αβ-equistatistical convergence of order γ.

Filter exhaustiveness and F-α-convergence of function sequences

In this work, we generalize the concepts of exhaustiveness, α-convergence, Cauchy sequence, pointwise convergence and uniform convergence for sequences of functions on metric spaces in terms of filters. We investigate some properties of these new concepts. We also examine the relations between the new concepts and the classical concepts.

Non-natural topologies on spaces of holomorphic functions

It is shown that every proper Frechet space with weak*-separable dual admits uncountably many inequivalent Frechet topologies. This applies, in particular, to spaces of holomorphic functions so solving in the negative a problem of Jarnicki and Pflug. For this case an example with a short self-contained access is added. It is a well known and often used fact, following from the closed graph theorem, that for any Frechet space of continuous functions the following is true: if any convergent sequence of functions in E also converges pointwise (or locally in L1) then convergence in E implies uniform convergence on compact sets. This is of particular interest if E is a Frechet space of holomorphic functions (see Krantz [2]). In this connection the question has been raised whether this might be true for any Frechet space E of holomorphic functions (see Jarnicki and Pflug [1, Remark 1.10.6, (b), p. 66]), that is, if every convergent sequence in E converges uniformly on compact sets. In the present note this question is solved in the negative in a very strict sense. For functional analytic tools and unexplained notation see [3]. The author thanks Peter Pflug for drawing his attention to this problem. Lemma 1 All proper (that is: not Banach) Frechet spaces with weak*-separable dual are linearly isomorphic. Proof: By dimE we denote the linear dimension of a linear space E and we set ω := CN with the product topology. By Eidelheit’s Theorem we know that there is a linear surjective map from E onto ω (see [3, 26.28]). This implies that dimω ≤ dimE. Let {y1, y2, . . . } be a weak*-dense set in E′. Then x 7→ (y1(x), y2(x), . . . ) is a linear, injective map E ↪→ ω. Therefore dimE ≤ dimω.

On the $\mathfrak I$-$R_0$ spaces and $\mathfrak I$-strong quasi-uniform convergence

Using ideals, we introduce a new class of topological spaces that are more general concept than $R_0$ spaces~{AC}. Different properties will be studied for that class of topological spaces. We also define a new kind of convergence for sequences of functions and formulate more general results to those contained in~{\cite{AS}}.

λ-Statistical Convergence of Sequences of Functions in Intuitionistic Fuzzy Normed Spaces

We studyλ-statistically convergent sequences of functions in intuitionistic fuzzy normed spaces. We define concept ofλ-statistical pointwise convergence andλ-statistical uniform convergence in intuitionistic fuzzy normed spaces and we give some basic properties of these concepts.

Generalized equi-statistical convergence of positive linear operators and associated approximation theorems

The concepts of equi-statistical convergence, statistical pointwise convergence and statistical uniform convergence for sequences of functions were introduced recently by Balcerzak et al. [M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007) 715–729]. In this paper, we use the notion of λ-statistical convergence in order to generalize these concepts. We establish some inclusion relations between them. We apply our new notion of λ-equi-statistical convergence to prove a Korovkin type approximation theorem and we show that our theorem is a non-trivial extension of some well-known Korovkin type approximation theorems. Finally, we prove a Voronovskaja type approximation theorem via the concept of λ-equi-statistical convergence. Some interesting examples are also displayed here in support of our definitions and results.

Equi-statistical convergence of positive linear operators

Balcerzak, Dems and Komisarski [M. Balcerzak, K. Dems, A. Komisarski, Statistical convergence and ideal convergence for sequences of functions, J. Math. Anal. Appl. 328 (2007) 715–729] have recently introduced the notion of equi-statistical convergence which is stronger than the statistical uniform convergence. In this paper we study its use in the Korovkin-type approximation theory. Then, we construct an example such that our new approximation result works but its classical and statistical cases do not work. We also compute the rates of equi-statistical convergence of sequences of positive linear operators. Furthermore, we obtain a Voronovskaya-type theorem in the equi-statistical sense for a sequence of positive linear operators constructed by means of the Bernstein polynomials.

Statistical convergence and ideal convergence for sequences of functions

Let I ⊂ P ( N ) stand for an ideal containing finite sets. We discuss various kinds of statistical convergence and I -convergence for sequences of functions with values in R or in a metric space. For real valued measurable functions defined on a measure space ( X , M , μ ) , we obtain a statistical version of the Egorov theorem (when μ ( X ) < ∞ ). We show that, in its assertion, equi-statistical convergence on a big set cannot be replaced by uniform statistical convergence. Also, we consider statistical convergence in measure and I -convergence in measure, with some consequences of the Riesz theorem. We prove that outer and inner statistical convergences in measure (for sequences of measurable functions) are equivalent if the measure is finite.

Remarks on transfinite sequences of functions that preserve convergence

In the paper [6] the author investigates sequences of functions (fn)n=1∞ preserving the convergence of sequences which are alternatively called conservative sequences of functions. In this note we extend this concept to the transfinite sequences of functions and prove that in this case the property of being conservative is equivalent to the property of being pointwise convergent. Then we introduce the concept of p-locally convergence for transfinite sequences of functions and show that if X is a locally Lindelöf first-countable T1-space and a transfinite sequence {fξ : X → R}ξ&lt;Ω converges p-locally uniformly to a function f : X → R, then C(f) = Lim C(fξ) where C(f) denotes the set of all continuity points of f.

Convergence for Sequences of Functions and an Egorov Type Theorem

For every ordinal ξ<ω1 we define a new type of convergence for sequences of functions (ξ-uniform pointwise) which is intermediate between uniform and pointwise convergence. Using this type of convergence we obtain an Egorov type theorem for sequences of measurable functions.